Q105 / 160 · TS EAMCET 2020 (14 Sep Shift 2) · Mathematics

If $f(x)=\frac{1}{x^3} \int_5^x\left(2 u^2-u f^{\prime}(u) d u\right.$, then $f^{\prime}(5)=$

  • A
    $\frac{13}{2}$
  • B
    $\frac{2}{13}$
  • C
    $\frac{13}{5}$
  • Correct answer
    D
    $\frac{5}{13}$

Solution

Given, $\begin{aligned} f(x) & =\frac{1}{x^3} \int_5^x\left(2 u^2-u f^{\prime}(u) d u\right. \\ \Rightarrow \quad x^3 f(x) & =\int_5^x\left(2 u^2-u f^{\prime}(u)\right) d u \end{aligned}$ On differentiating w.r.t ' $x$ ', $\begin{aligned} x^3 f^{\prime}(x)+3 x^2 f(x) & =2 x^2-x f^{\prime}(x) \\ f^{\prime}(x) & =\frac{2 x^2-3 x^2 f(x)}{x^3+x} \\ f^{\prime}(5) & =\frac{2(5)^2-3(5)^2 f(5)}{5^3+5} \\ f^{\prime}(5) & =\frac{50-0}{130}=\frac{50}{130} \quad[\because f(5)=0] \\ f^{\prime}(5) & =\frac{5}{13} \end{aligned}$

Hence, the correct answer is option: 'D'.

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